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Against Deductive Closure

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Against Deductive Closure Against Deductive Closure Abstract: The present article illustrates a conflict between the claim that rational belief sets are closed under deductive consequences, and a very inclusive claim about the factors that are sufficient to determine whether it is rational to believe respective propositions. Inasmuch as it is implausible to hold that the factors listed here are insufficient to determine whether it is rational to believe respective propositions, we have good reason to deny that rational belief sets are closed under deductive consequences. Key words: deductive closure, rational belief, full and partial belief 1. Introduction Within formal theories of rational belief, it is often assumed that rational belief sets are closed under deductive consequences (e.g., within epistemic logic and within the dominant AGM theory of belief revision).1 The ideal of believing all deductive consequences of the propositions one believes is one that human agents cannot hope to satisfy. For this reason, proponents of formal theories that assume closure generally maintain that the theories properly apply to agents with unlimited deductive abilities, and acknowledge that the prescriptive force of the theories is limited in some respects, in the case of (finite) human beings. In other words, proponents of deductive closure generally adopt the view that deductive closure is an ideal that, nonetheless, exerts some prescriptive force on human beings. This view has some intuitive plausibility, and is supported by common epistemic practice, which appears to presuppose that rational belief tracks logical implications. For example, we routinely assume that rational agents are committed to believe the logical The final publication is available at Wiley via: http://onlinelibrary.wiley.com/doi/10.1111/theo.12103/full 1 The canonical sources are (Hintika 1962) and (Alchourrón, Gärdenfors, & Makinson 1985). 2 consequences of propositions that they believe, with the consequence that we can compel an agent to repudiate some element of a set of believed propositions by demonstrating that those propositions entail another proposition that the agent is unwilling to accept (Kaplan 1996, 96- 7). We also routinely assume that we can establish the correctness of believing a proposition by showing that it follows from other propositions that we are justified in believing (Pollock 1983, 247). Inasmuch as the described epistemic practices ‘seem right’, they provide intuitive support for the claim that (ideal) rational belief sets are closed under deductive consequences. Despite its intuitive plausibility, the claim that rational belief sets are closed under deductive consequences is a subject of ongoing debate. The present article is motivated by the failure of what was once regarded as a relatively good argument against the claim that rational belief sets are closed under deductive consequences.2 This (failed) argument derives from the apparent fact that high rational personal probability is a necessary condition for rational belief, and from the fact that degree of probability is not generally preserved when we aggregate propositions. In combination, these two facts suggest that the conjunction of a set of propositions whose elements are each probable, and objects of rational belief, may be improbable, and therefore not an object of rational belief. So, according to the argument: Given a set of propositions whose elements are each objects of rational belief, it is not the case, in general, that their conjunction will also be an object of rational belief. An advocate of deductive closure can evade the present argument, by maintaining that a personal probability of one is a necessary condition for rational belief, noting that probability one is preserved when we aggregate propositions. But the claim that it is only rational to believe a proposition if its probability is one is extreme, and incorrect as a descriptive claim about the semantics of belief ascriptions. 2 I articulated the following argument in an unpublished paper (Thorn ms.) that was presented at several conferences in 2006. A similar argument appears in (Foley 2009). 3 While the preceding argument against deductive closure may once have been persuasive, its invalidity and ultimate failure is now evident, inasmuch as Leitgeb (2013, 2014) has demonstrated the ‘formal possibility’ of relating rational personal probability to rational belief, in such a way that the following two theses are satisfied (where r is a constant, such that 0< r < 1): (DC) Deductive Closure: Rational belief sets are closed under deductive consequences. (LT) The Lockean Thesis, left to right: Having a rational personal probability of at least r is a necessary condition for rational belief (cf. Foley 1993). Just what value r has will be of small consequence here, since the problems with (DC) are generated irrespective of r’s value. As such, I will treat r as a numeric constant, and leave the question open as to what value it would have within a true and maximally informative instance of (LT). Given the preceding, my commitment to (LT) may be understood as an existential claim: There exists an s (0 < s < 1), such that for all agents, a rational probability of at least s is necessary for rational belief. Faced with the failure of the argument described above, I will here endeavor to provide a new and better argument against (DC). In particular, I aim to show (in the context of some other reasonable assumptions, including (LT)) that (DC) conflicts with a very inclusive claim about the factors that are sufficient to determine the rationality of believing respective propositions. From the outset, it is important to observe that while Leitgeb’s theory is the catalyst for the argument presented here, his theory is not the primary or exclusive target of the argument. Inasmuch as Leitgeb’s theory maintains (DC) (along with (LT)), the theory lies within the argument’s target area. But Leitgeb’s theory is rather ambitious, aiming to uphold (DC) along with both directions of a ‘context dependent’ version of the Lockean Thesis. 4 Before considering the connection between Leitgeb’s view and (LT), it is useful to compare (LT ) with the right-to-left direction of the Lockean Thesis: (LT) The Lockean Thesis, right to left: Having a rational personal probability of at least r is a sufficient condition for rational belief. While many find (LT) plausible (including myself), the claim is subject to reasonable doubts. Motivated by Kyburg’s Lottery Paradox (1961) and the desire to uphold the claim that rational belief sets are consistent, Levi (1967) and others rejected (LT), proposing that practical considerations, and/or considerations of expected veritistic value (or epistemic utility), could have a bearing on what rational personal probability threshold was sufficient to license rational belief (cf. Lehrer 1975).3 While Levi’s rejection of (LT) may have been motivated by the Lottery Paradox, grounds for rejecting (LT) have independent plausibility. If we regard rational personal probability as a function of the strength of an agent’s evidence, then it is not implausible to think that the standard of evidence sufficient to license rational belief could vary according to context. Perhaps the evidential standards specific to believing a proposition depend on its subject matter (e.g., mathematical, metaphysical, or mundane) and/or the stakes involved in case the belief is in error. Once one allows such contextual variability in evidential standards, it is not implausible to deny that there is any rational personal probability threshold (excluding r = 1) that is sufficient for rational belief. Moreover, if one holds that rational belief sets are consistent, then (by consideration of lottery cases) one will probably want to deny that there is any rational personal probability threshold (excluding r = 1) that is (in all contexts) sufficient for rational belief. Unlike (LT), (LT) is about as secure as any conceptual claim (outside of logic and mathematics) could be. It is nigh absurd to claim that an agent may rationally believe some proposition in the case where the agent’s rational personal probability for that proposition is 3 Makinson’s Preface Paradox (1965) was also important in raising suspicions about deductive closure. 5 less than 0.5 (assuming an epistemic conception of rationality). If one is not convinced that a rational personal probability of 0.5 is necessary for rational belief, then suppose that r is less than 0.5. Notice that the version of (LT) presented here is rather modest, demanding only that r > 0. Up until now, I have assumed a ‘context independent’ reading of the Lockean Thesis, which has the following consequence: There exists some s (0 < s < 1), such that, for all agents, a rational probability of at least s is necessary and sufficient for rational belief. Leitgeb, himself, avoids the Lottery Paradox (understood as a problem for (LT)) by defending a (far weaker) ‘context dependent’ variant of the Lockean Thesis: For each agent (in the context specific to that agent), there is some value s (0 < s < 1), such that a rational probability of at least s is necessary and sufficient for rational belief.4,5 Taken individually, each direction of the context dependent Lockean thesis is plausible. The context dependent version of (LT) is no less plausible than the (near indubitable) context independent version of (LT). It is also noteworthy that the context dependent version of (LT) all but implies the context independent version of (LT). If, for each agent, there is a context dependent minimum probability that is necessary for rational belief, then there must be a greatest lower bound on the set of such thresholds. Such a lower bound would then constitute a context independent minimum probability for rational belief. Given the more or less indubitable assumption that this lower bound is greater than zero, the context dependent version of (LT) implies the context independent version. This means that my argument against (DC), by appeal to the context independent version of (LT), could just as well have appealed to the context dependent version of (LT).
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